graph layouts by t/sne - university of arizonakobourov/tsne-eurovis17.pdf · j. f. kruiger et al. /...

Eurographics Conference on Visualization (EuroVis) 2017J. Heer, T. Ropinski, and J. van Wijk(Guest Editors)

Volume 36 (2017), Number 3

Graph Layouts by t-SNE

J. F. Kruiger1,2, P. E. Rauber1,3, R. M. Martins4, A. Kerren4, S. Kobourov5, A. C. Telea1

1University of Groningen, the Netherlands2École Nationale de l’Aviation Civile, France

3University of Campinas, Brazil4Linnaeus University, Sweden5University of Arizona, USA

AbstractWe propose a new graph layout method based on a modification of the t-distributed Stochastic Neighbor Embedding (t-SNE)dimensionality reduction technique. Although t-SNE is one of the best techniques for visualizing high-dimensional data as 2Dscatterplots, t-SNE has not been used in the context of classical graph layout. We propose a new graph layout method, tsNET,based on representing a graph with a distance matrix, which together with a modified t-SNE cost function results in desirablelayouts. We evaluate our method by a formal comparison with state-of-the-art methods, both visually and via established qualitymetrics on a comprehensive benchmark, containing real-world and synthetic graphs. As evidenced by the quality metrics andvisual inspection, tsNET produces excellent layouts.

1. Introduction

Graph drawing (GD) is an important subfield of information visu-alization. Yet, making a ‘good’ drawing becomes more difficult forlarge and densely-connected graphs. Assuming edges are drawn asstraight-line segments, the main goal of graph drawing is assign-ing appropriate coordinates to graph nodes, an operation knownas graph layout. Many graph layout methods exist, e.g., force-directed methods [Ead84,FR91], spectral methods [Hal70,BP07],and dimensionality-reduction methods [KS80, HK04, GKN05].

In contrast to traditional GD methods, dimensionality reduction(DR) methods typically map the graph connectivity informationto a high-dimensional space [HK04], e.g., via the graph-theoreticdistances between all pairs of nodes [KS80, KKH89]. This high-dimensional data is then suitably mapped to 2D space to create thegraph layout. While graph drawing by DR is elegant and simple,few methods exist in this class. This provides opportunities for im-proved graph layout methods that take advantage of advances inDR which provide increasingly more accurate, flexible, scalable,and easy-to-use techniques, as described in detail in Sect. 2.

In this paper, we propose a new DR-based method for graphdrawing. We leverage t-SNE [vdMH08], one of the most acclaimedDR methods, extensively used in machine learning, computer vi-sion, and information visualization [MKS∗15, DJV∗14]. In con-trast to other DR methods used for GD—which aim to preservedistances between points when mapping these to 2D space—t-SNEaims to preserve point neighborhoods. As we argue later on, neigh-borhood preservation is desirable for GD. We describe a way to

encode the graph as a distance matrix, along with a suitable modifi-cation to the standard t-SNE cost function and iteration parameters,that allow us to map the graph’s connectivity to a 2D projection.Our method—called tsNET, a play on ‘t-SNE’ and ‘network’—issimple to implement and use, and has a single free parameter, per-plexity, described in more detail in Sect. 5. Moreover, tsNET inher-its the proven robustness and quality properties of t-SNE, and gen-erates high quality layouts for a wide variety of graphs. We evaluatetsNET by comparing it quantitatively and qualitatively with severalstate-of-the-art methods on a representative family of real-worldand synthetic graphs.

This paper is organized as follows. Sect. 2 discusses related workin DR and graph layouts. Sect. 3 presents our method. Sect. 4 showsour results on a comprehensive graph benchmark. Sect. 5 discussestsNET, and Sect. 6 draws conclusions and outlines future work.

2. Background

Our work is related to both dimensionality reduction and graphdrawing. We review related work in both of these areas below.

2.1. Dimensionality reduction

Dimensionality reduction (DR) considers the problem of reduc-ing the number of dimensions of a dataset while retaining rel-evant data patterns. Given a set of N n-dimensional observa-tions {xi ∈ Rn}N

i=1, a DR technique is a function f : Rn → Rm

that maps {xi} to a set of low-dimensional points {yi ∈ Rm}Ni=1,

c© 2017 The Author(s)Computer Graphics Forum c© 2017 The Eurographics Association and JohnWiley & Sons Ltd. Published by John Wiley & Sons Ltd.

J. F. Kruiger et al. / Graph Layouts by t-SNE

where m� n and typically m ∈ {2,3}, so that the so-called ‘datastructure’ is preserved [MCMT14]. DR methods can be classi-fied into distance-preserving and neighborhood-preserving meth-ods. Distance-preserving methods aim to minimize a cost such asthe so-called aggregated normalized stress

σ = ∑i, j

(d(xi,x j)−‖yi−y j‖

d(xi,x j)

)2

. (1)

Here, d(·, ·) is a distance metric over the input space of f and ‖ · ‖is usually the Euclidean 2D distance. Such methods are used whenreasoning about inter-point distance ratios is important.

Neighborhood-preserving methods try to maximize overlap be-tween k-nearest neighborhoods in the input (Rn) and output (Rm)spaces, and help find groups and outliers. Distance-preserving andneighborhood-preserving methods have related, but not identical,aims. In particular, distance-preservation gets hard when the di-mensionality n is high [dST03,JPC∗11]; neighborhood-preservingmethods handle high n values better [vdMH08, PLvdM∗15].

DR methods can also be classified into projections or distance-based methods. Projections require the actual Rn coordinates ofthe N input points, and thus require O(nN) in space complex-ity [MCMT14]. Such methods are not directly applicable in a GDsetting, as we do not typically have a way to generate n-dimensionalcoordinates of nodes in a graph. Distance-based methods, alsocalled multidimensional scaling (MDS) methods, use only the pair-wise distances between all xi, and require O(N2) space complex-ity [Tor52, BSL∗08]. MDS methods are more general than projec-tions, as they do not require explicit Rn coordinates for the inputpoints.

In the MDS class, ISOMAP [TdSL00] first determines whichdata points are neighbors in Rn, then determines geodesic distancesbetween these points (in Rn), and finally performs classical MDSwith the pairwise geodesic distances. LSP [PNML08] works byfirst projecting a subset of the Rn points via classical MDS, andnext uses neighborhood information from Rn to place the rest ofthe data points in Rm. LAMP [JPC∗11] is similar to LSP. It firstprojects a subset of points via classical MDS, yet it also allowsthe user to interact with the positioning of these points. IDMAP[MPd06] uses a distance metric based on cosine similarity, and sub-sequently makes a FastMap [FL95] projection which in turn usesa force-based placement approach. Landmark MDS [dST03] andPivotMDS (PMDS) [BP07] are sampling-based approximations ofclassical MDS. They place representatives from the input data withhigh accuracy, while remaining nodes are placed via linear combi-nations of the representatives. Both methods are very fast. PMDStypically produces better results as it takes into account distancesto non-representatives when placing the representatives. For a com-plete overview of DR methods, we refer the reader to recent surveysin the area [SVPM14, CG15].

2.2. Graph layouts

General considerations: A graph G = (V,E), also known as a net-work or relational dataset, consists of a set of nodes V = {xi}N

i=1and a set of edges E =

{(xi,x j)

}⊆ V ×V that map relations be-

tween nodes in V . Both nodes and edges can have additional data

attributes (such as weights), although many GD methods ignoreweights.

A graph layout algorithm can be seen as a function L that takesas input a graph G and outputs a set of (typically 2D) node coor-dinates, i.e., L(G) = {yi ∈ R2}N

i=1. We cannot describe all graphlayout techniques here; for recent comprehensive surveys on thetopic we refer to [Tam13,vLKS∗11]. Next we provide details aboutgraph layout methods that are particularly relevant for our goal—drawing large and potentially highly connected graphs.

A popular choice for graph layouts is SFDP [Hu05], which isavailable in GraphViz [ATT]. SFDP uses a multilevel approachand quadtree optimizations, which makes it usable for fairly largegraphs. A drawback of SFDP is that it rigidly enforces uniform-length edges, which can be aesthetically pleasing but does not al-ways result in the most insightful layout.

GRIP [GK01] is a nearly linear multilevel method that uses nodefiltrations based on maximal independent sets. A similar methoduses a hierarchy of progressively less coarse graphs [Wal01]. Mul-tilevel approaches make graph layout more scalable, but assumethat pairs of nodes with small graph-theoretic distances also havesmall distances in the optimal layout [Noa07a].

The r-PolyLog class of energy models [Noa07a], which containsthe LinLog energy model, aim to compromise between enforcinguniform edge-lengths and separating highly connected nodeclusters. LinLog can draw graphs that consist primarily of clustersvery well, but is less successful for other graph types.

Graph layouts by DR: A particular class of graph layout meth-ods is explicitly based on DR techniques and we discuss these inmore detail, as they are most relevant to our method. A key to theDR-approach is suitably defining the distance d so that minimiz-ing the stress σ (Eqn. (1)) reflects a good layout. For this, d can beset to the graph-theoretical shortest-path distance between nodes xiand x j. Minimizing stress for graph drawing was originally doneby [KS80], and later by [KKH89]. NEATO [Nor04] is a popu-lar implementation that minimizes the stress, and is available inGraphViz [ATT]. Graph drawing by stress majorization [GKN05]minimizes stress by means of majorization. A drawback of min-imizing stress directly is that it fails to untangle many complexgraphs because it aims to rigidly preserve distance.

ACE [KCH03] is a very fast method that uses a matrix-representation for the graph and several of its properties. It usestransformations to coarser, lower-dimensional matrices, and findsa layout that is projected back to the higher-dimensional repre-sentation where it is adjusted to solve the original problem. ACEworks well for mesh-like graphs, but produces sub-optimal layoutsfor sparsely-connected graphs, such as trees. A related multilevelmethod is [FT07].

Graph drawing by high-dimensional embedding [HK04] is avery fast method that first makes an n-dimensional embedding, us-ing n pivot nodes and the graph-theoretic distances from those pivotnodes to all other nodes. Subsequently, dimensionality is reducedfrom n to 2 by means of PCA. This method does not work wellfor non-mesh-like graphs. Also, PCA is well-known to poorly pre-

c© 2017 The Author(s)Computer Graphics Forum c© 2017 The Eurographics Association and John Wiley & Sons Ltd.


serve distances when the n-dimensional data does not lie on a linearsubspace of Rn.

[Kor04] builds on the high-dimensional embedding idea by ad-ditionally considering the high-dimensional subspace spanned bythe eigenvectors of the Laplacian matrix of the graph, and projectsthis to 2D utilizing the graph’s structure. This method works wellfor meshes but less so for other graph types.

ws-SNE [YPK14] proposes a general framework to theoreticallyunify DR methods and GD methods. ws-SNE also uses cost termsoriginating from t-SNE, but employs different distance measuresand optimization strategies than we propose here (for details, seeSect. 3.2). More recently, s-SNE [LYC16] built further on ws-SNEby considering spherical projections. We did not incorporate thesemethods to our comparison due to the minimal implementation de-tails of ws-SNE and the different layout space of s-SNE.

2.3. Contributions

Compared to other DR-based graph layout methods, our proposalis characterized by the following three aspects:

• we use the neighborhood-preserving t-SNE technique rather thanthe distance-preserving techniques prevalent in current methods;• we use a modified cost function with terms based on t-SNE and

classical force-based methods;• we use the graph-theoretical shortest-path distances in the input

space.

Additionally, we propose a refined method, tsNET*, which im-proves over tsNET by using suitable node pre-placement. Finally,a major contribution of our work is the extensive evaluation of sev-eral graph layout methods on a varied set of graphs.

3. Method

We next present our graph layout method. As our method exploitst-SNE, we first outline t-SNE (Sect. 3.1). Our method proper fol-lows in Sect. 3.2.

3.1. t-SNE

t-SNE is a neighborhood-preserving MDS method that is often usedfor creating scatterplots of high-dimensional data [vdMH08]. Itworks by defining probabilities pi j of picking a point-pair in theinput space and probabilities qi j of picking a point-pair in the out-put (2D) space. The probability pi j of picking the pair (xi,x j) is thesymmetrized version of the conditional probabilities pi| j and p j|i

pi j = p ji =pi| j + p j|i

2N, pii = 0,

where the conditional probabilities p j|i are given by the normalizedGaussian distribution

p j|i = exp

(−

d2i j

2σ2i

)/∑k

k 6=i

exp

(− d2

ik2σ2

i

), pi|i = 0,

which gives the probability that xi has x j as neighbor. Here, di j isthe distance between xi and x j in input space. The Gaussian stan-dard deviation σi can be either hand-picked or found e.g., by binary

search so that the perplexity κi = 2−∑

jp j|i log2 p j|i

for every point ximatches a user-defined value. The perplexity measures the effectivenumber of neighbors of a point. If xi is in a dense region, σi gener-ally has a low value to match the perplexity. In sparser regions, σitypically gets a large value. Finally, qi j, the probability of pickingthe pair (yi,y j) in the output (2D) space, is given by

qi j = q ji =(1+

∥∥yi−y j∥∥2)−1

∑k,lk 6=l

(1+‖yk−yl‖2)−1, qii = 0,

which is a normalized (heavy-tailed) Student’s t-distribution.

Node positions yi in the 2D space are found by minimizing (withrespect to yi) the Kullback-Leibler divergence between the proba-bilities of picking pairs of low- and high-dimensional data points:

CKL = ∑i, ji6= j

pi j logpi j

qi j.

This minimization is typically done using gradient descent. Thegradient ∂CKL

∂yiis easily expressed in analytic form and the mini-

mization is therefore straightforward to implement.

t-SNE is among the best MDS methods that succeeds in em-phasizing relevant point-groups while also capturing local variationwithin groups [MKS∗15, DJV∗14].

3.2. tsNET

Given a graph G, our layout method has three steps, as follows.We first compute the graph-theoretic shortest-path distances di j be-tween all nodes xi and x j of G. If no path exists between xi and x j,di j is set to a large arbitrary value. This yields a symmetric N×Nmatrix D = (di j). Next, we use D to construct a cost function C asa sum of three terms:

C = λKLCKL +λc

2N ∑i‖yi‖2− λr

2N2 ∑i, ji 6= j

log(∥∥yi−y j

∥∥+ εr). (2)

Here, term 1 is the Kullback-Leibler divergence from t-SNE(Sect. 3.1); term 2 denotes so-called compression, which is knownto reduce the t-SNE optimization time [vdMH08]; and term 3 re-pulses nodes yi to prevent undesirable clutter and attain even dis-persion (entropy in [GHN13]). The weights λKL, λc, and λr governthe impacts of the different terms, and do change during optimiza-tion. A small regularization constant εr =

120 is added to treat sit-

uations of nearly co-located nodes. We have also considered elec-trostatic repulsion, but finally settled on the entropy from [GHN13]motivated by empirical results and the simple analytic form of itsgradient when εr = 0: ∂

∂yk( 1

2 ∑i, ji6= j

log(∥∥yi−y j

∥∥)) = ∑i

i 6=k

yk−yi

‖yk−yi‖2 .

The 2D node positions {yi}Ni=1 are found by minimizing C by

means of momentum-based gradient descent, as in classical t-SNE.For this, we propose a three-stage method. First, we randomly ini-tialize yi. Secondly, we do momentum-based gradient descent opti-mization on C with (λKL,λc,λr) = (1,1.2,0) until the summed dis-placement of nodes in the layout is smaller than a given threshold.



In the third stage, we use the node positions to do the same opti-mization, but with (λKL,λc,λr) = (1,0.01,0.6). The second stageaims at accelerating the ‘untangling’ of the graph with the aid ofcompression. The third stage optimizes for the final layout wherewe want to prevent too closely placed vertices and the fisheye dis-tortion introduced by high λc. We call the above method tsNET.

As a refinement of this method, we initialize yi using Pivot-MDS in stage 1, and then in stage 2 use the parameter settings(λKL,λc,λr) = (1,0.1,0). Stage 3 remains unchanged. The weightof the compression term in stage 2 has been lowered to prevent itfrom distorting the PivotMDS layout. We call this method tsNET*.Section 4 will show the advantages of tsNET* over tsNET.

To better explain the effect of the individual cost terms, Fig. 1shows layouts computed with and without certain cost terms ona graph of 1005 nodes and 3808 edges. We see that plain t-SNE(λc = λr = 0), shown in Fig. 1a, generates a layout with unde-sired occlusions, folds, an uneven node density, and has a slowconvergence (requires many iterations). As such, plain t-SNE isnot enough to generate a good graph drawing. Such aspects areremoved by the additional components of tsNET and tsNET*, asFigs. 1b to 1e show. We observed similar effects on other graphswe tested (which are introduced further in Sect. 4.1).

(a) Plain t-SNE:1027 iterations

(b) t-SNE + earlycompression:

1167 iterations

(c) t-SNE + laterepulsion:

1387 iterations

(d) tsNET:522 iterations

(e) tsNET*:307 iterations

Figure 1: The effect of the additional cost terms, with the numberof gradient descent iterations needed for convergence.

Important differences exist between ws-SNE and tsNET. First,ws-SNE uses different cost functions: Different repulsion termsare used (compare Eqn. (2), term 3, with ∑i j pi j

∥∥yi−y j∥∥2

+

λ∑i j Mi j exp(−∥∥yi−y j

∥∥2) from [YPK14]). We use a compression

term (Eqn. (2), term 2) to reduce optimization time. To our un-derstanding, ws-SNE does not (and cannot) use such a term, as itrelies on so-called separable divergences (sums of pairwise termsdepending on data points i and j). Moreover, ws-SNE uses the de-gree centrality of nodes in its repulsion term (Mi j in the above ex-pression), to favor good placement of central nodes. As shown nextin Sect. 4, we obtain good results even without computing such aterm, which can be expensive for large graphs. Finally, we compute

pi j from shortest-path distances, whereas ws-SNE assumes that pi jare computed from the k-nearest-neighbors of graph nodes; how-ever, how this is precisely done is not specified in [YPK14], nor areshortest-path distances named in this context.

We have implemented tsNET in Python 3, using the Theano nu-merical library [BBB∗]. Our open source is available online at[Kru16]. Its space and time complexities are O (N2) and O (tN2),resp., with t the number of gradient descent iterations—around sev-eral hundreds in our benchmark. If desired, massive accelerationsare easily doable, see further Sect. 5.

4. Evaluation

We evaluate our layout method by comparing it to several state-of-the-art layout methods on a set of synthetic and real-world graphswith various sizes (Sect. 4.1). Sect. 4.2 compares the obtainedgraph layouts visually. Sects. 4.3 to 4.5 compare layouts via sev-eral quantitative metrics. Sect. 4.6 compares running times for ourmethods. Sect. 4.7 shows how our layouts can be further enhancedby edge bundling. Finally, Sect. 4.8 demonstrates that our methodcan also construct 3D layouts.

4.1. Benchmark

To evaluate tsNET(*), we used a wide set of graphs and layoutmethods, as follows.

Tab. 1 lists the graphs we used for benchmarking. These includereal-world collaboration networks, structural problems from theHarwell-Boeing collection [DGL89a], mesh-like graphs, and tree-like graphs. Many of our graphs come from the Florida collection[DH11], a well-known source for graph drawing and graph analy-sis. Other sources are synthetic graphs computed with [Pei14,ATT]and graphs from [MAH∗12] (a recent DR-based graph layoutmethod). These graphs are used in additional papers beyond thoselisted in Tab. 1. However, these could not all be included for spacereasons.

We compare tsNET(*) with the following layout methods:

IDMAP [MPd06], as implemented in VisPipeline [POM07];

LSP [PNML08], as implemented in VisPipeline [POM07], withnumber of control points set to 10% of the number of nodes;

PMDS [BP07], as implemented in Tulip 4.8.1 [Aub04], with num-ber of pivots set to 10% of the number of nodes;

SFDP [Hu05], as implemented in GraphViz 2.38 [ATT];

LinLog [Noa07a], as implemented by the author [Noa07b];

GRIP [GK01], as implemented by the authors [Yus01]. The GRIPimplementation in Tulip was also evaluated, but it produced visu-ally inferior results, which are not shown here.

NEATO [Nor04], as implemented in GraphViz 2.38 [ATT];

All compared methods are well-known, and have open-source im-plementations, which makes it easy to replicate our experiments.We chose IDMAP and LSP, which are DR projections, specificallyto show the advantages of our t-SNE-based method as opposed toother DR methods. We chose GRIP, PMDS, SFDP, and LinLogsince these are well-known methods, present in many graph draw-ing papers. However, it is fair to note that these methods do not



Name Type |V | |E| Source Also seen indwt_72 planar, structural 72 75 [DGL89b] [HBFR14]lesmis collaboration network 77 254 [Knu93]can_96 mesh-like, structural 96 336 [DGL89b] [HBFR14]rajat11 miscellaneous 135 377 [Raj]jazz collaboration network 198 2742 [GD03]visbrazil collaboration network 222 336 [MAH∗12]grid17 planar, mesh-like, structural 289 544 synthesized using [Pei14]mesh3e1 mesh-like, structural 289 800 [NAS] [Tam13]netscience collaboration network 379 914 [New06] [GJ12]dwt_419 structural 419 1572 [DGL89b] [PV13, HBFR14]price_1000 planar, tree 1000 999 synthesized using [Pei14]dwt_1005 structural 1005 3808 [DGL89b] [OKB16]cage8 miscellaneous 1015 4994 [VHBB02]bcsstk09 mesh-like, structural 1083 8677 [DGL89b] [FT07]block_2000 clusters 2000 9912 synthesized using [Pei14]sierpinski3d structural 2050 6144 synthesized using [ATT] [GK01]CA-GrQc collaboration network 4158 13422 [New01] [BG13]EVA collaboration network 4475 4652 [NLGC02]3elt planar, mesh-like 4720 13722 [DP] [Wal01, OKB16, FT07]us_powergrid structural 4941 6594 [WS98] [OKB16, GHK13]

Table 1: Types and sizes of graphs in our benchmark. All graphs we used are publicly available, as indicated.

use a ‘full stress’ model, in contrast to ours. On the other hand, wecompare also with NEATO, which does use such a model. For allgraphs, we use the same method parameters, as given in Sect. 3.2,except λr = 0.1 for dwt_72, which gave better results.

4.2. Visual comparison

Tabs. 5 and 6 visually compare the obtained layouts; rows indi-cate graphs, and columns indicate layout methods. We see thattsNET* is of the same or, we argue, in some cases higher visualquality than the compared layouts, in terms of node clutter, mini-mizing unnecessary distortions, and preserving symmetry in the in-put graphs. Edge length distribution, encoded in edge color, showsthat tsNET* slightly favors short edges (column 2 in Tabs. 5 and 6shows relatively more red than the other columns). Yet, this doesnot adversely affect the visual quality of the layouts. Another in-teresting point is that tsNET* has a wider edge-length distributionthan most other layouts. For instance, for dwt_1005, block_2000and 3elt, our method chooses to draw a few long edges (blue) butkeeps most other edges short (red). This selectively ‘pulls apart’ afew connected nodes so that the layout is nicely spread out overthe 2D plane. Another finding is that our layouts appear moresmooth and organic than others; structurally-similar graph regions,e.g., in mesh-like graphs (dwt_1005, bcsstk09, can_96, grid17,mesh3e1, and dwt_419) also appear structurally similar in our lay-outs, whereas some of the other methods produce unnecessary de-formations. Finally, we see that tsNET* improves upon tsNET byremoving artifacts where part of the tsNET layout is ‘folded over’,see e.g., can_96, dwt_1005 and 3elt.

4.3. Normalized stress metric

A compact way to assess the distance-preservation of a layoutis to use the scalar normalized stress metric σ (Eqn. (1)), seee.g., [OKB16, GHN13]. Tab. 2 shows σ for all considered lay-outs. We see that tsNET(*) do not reach significantly better val-ues than all other considered methods. In particular, NEATO, GRIP

and IDMAP perform better here. Yet, if we look at the actual lay-outs (Tabs. 5 and 6), IDMAP and GRIP deliver arguably worseresults—lack of symmetry preservation (IDMAP on dwt_1005, bc-sstk09), weak cluster preservation (block_2000),unnecessary dis-tortions (3elt, IDMAP and GRIP on bcsstk09), and folding arti-facts (NEATO on dwt_419, dwt_1005). Specifically, consider thelow stress values of NEATO, IDMAP and GRIP on block_2000. Itis evident that the stress metric does not capture the quality aspectsthat we require for this graph. This makes us hypothesize that puredistance preservation might not be the ideal aim of a good graphlayout method. Rather, having groups of nodes topologically closein the graph be also geometrically close in the layout allows read-ing the drawing well [Tam13,Noa07a]. This neighbor preservation,discussed in Sect. 4.5, is precisely what tsNET aims at.

tsNET tsNET* IDMAP LSP PMDS SFDP LinLog GRIP NEATOdwt_72 0.048 0.048 0.039 0.118 0.072 0.061 0.201 0.038 0.043lesmis 0.109 0.111 0.111 0.226 0.162 0.112 0.219 0.099 0.084can_96 0.112 0.084 0.085 0.088 0.092 0.075 0.091 0.104 0.072rajat11 0.096 0.097 0.097 0.098 0.107 0.096 0.194 0.074 0.064jazz 0.127 0.128 0.126 0.155 0.158 0.137 0.387 0.114 0.110visbrazil 0.098 0.098 0.083 0.113 0.157 0.081 0.474 0.089 0.068grid17 0.021 0.021 0.016 0.026 0.025 0.023 0.210 0.018 0.014mesh3e1 0.014 0.014 0.004 0.006 0.003 0.036 0.076 0.009 0.005netscience 0.101 0.100 0.075 0.096 0.103 0.105 0.182 0.070 0.063dwt_419 0.024 0.024 0.023 0.022 0.026 0.052 0.112 0.022 0.054price_1000 0.165 0.160 0.117 0.159 0.242 0.133 0.190 0.126 0.093dwt_1005 0.152 0.035 0.030 0.030 0.029 0.029 0.219 0.026 0.096cage8 0.185 0.203 0.151 0.142 0.140 0.147 0.207 0.150 0.122bcsstk09 0.022 0.022 0.037 0.027 0.066 0.024 0.096 0.021 0.015block_2000 0.193 0.189 0.164 0.205 0.181 0.162 0.302 0.155 0.144sierpinski3d 0.077 0.093 0.152 0.092 0.091 0.079 0.310 0.068 0.063CA-GrQc 0.182 0.189 0.150 0.175 0.182 0.148 0.220 0.172 0.129EVA 0.171 0.161 0.148 0.141 0.233 0.124 0.325 0.149 0.0983elt 0.110 0.090 0.052 0.045 0.057 0.060 0.317 0.049 0.046us_powergrid 0.150 0.101 0.074 0.080 0.090 0.094 0.271 0.091 0.058average 0.103 0.094 0.083 0.097 0.106 0.085 0.219 0.078 0.069

Normalized stress (rel.)low (good) high (bad)

mean

Table 2: Normalized stress (Eqn. (1)). Layouts are uniformly scaledto give the minimal stress value. Cell colors encode the relativestress on the same row.

4.4. Distance preservation plots

Another measure used to evaluate the quality of a graph layout,or projection, is to look at the correlation between inter-point dis-



tances in input vs. output space [BP09, GHN13, JPC∗11]. Fig. 2shows such plots for our considered graphs and methods. Here,grayscale encodes the density of point-pairs (darker=more). Bothinput and output space distances are normalized by the largest dis-tances in the individual spaces. Such plots help see how well graph-theoretic distances are preserved in the layout—for a perfect preser-vation, the plot is a diagonal line. Note that the x-axis is discrete,since D contains graph-theoretic distances.

The plots for tsNET* do not show a larger spread (with respect tothe diagonal) than those of compared methods. This is very interest-ing, since our underlying t-SNE method does not aim to preservedistances, but neighborhoods, as explained in Sect. 3. Moreover,the absence of dark regions close to the x-axis for tsNET(*) indi-cates that our layouts have few false neighbors, i.e., nodes are wellspread out in the available 2D layout space. The gaps in the plots forblock_2000 by tsNET(*) and LinLog indicate that graph clustersare well separated. Yet, although LinLog can separate such clus-ters, it performs worse than our method for other types of graphs(see dwt_1005, 3elt and us_powergrid). Also, the cluster separationof LinLog is arguably of lower quality than ours, given the darkregions in the bottom-right part of its plot for block_2000, whichindicate false neighbors. Another, more general observation fromthe plots in Fig. 2 is that the distribution of points for tsNET(*)tends to be more symmetric with respect to the diagonal, i.e., thereis a balance between the number of point-pairs in the layout thatare false neighbors and those that are missing neighbors.

4.5. Neighborhood preservation metric

As argued in Sect. 4.3, we aim to provide a good graph layoutby preserving neighbors rather than distances. To measure this,we use a neighborhood preservation metric similar to [GHN13].While [GHN13] use a fixed neighborhood size, we change this sizeaccording to the position of a node in the graph. For every node xiin a graph G = (V,E), we define a neighborhood NG(xi,rG) as thenodes with a graph-theoretic distance of at most rG from xi, i.e.,

NG(xi,rG) ={

x j ∈V | di j ≤ rG}.

Likewise, we define an equally sized neighborhood of xi in thelayout NY (xi,ki) as the set of nodes corresponding to the pointsthat are the ki-nearest-neighbors of yi in the 2D layout space, withki = |NG(xi,rG)|. The neighborhood preservation ν is next definedas the fraction of nodes that are shared by these neighborhoods,averaged over all nodes in G, i.e.,

ν =1|V |∑i

|NG(xi,rG)∩NY (yi,ki)||NG(xi,rG)∪NY (yi,ki)|

.

This is the Jaccard similarity between the neighborhoods NG andNY , averaged over G. Simply put, ν tells how well the 2D (layout)neighborhoods match the input (graph) neighborhoods [MMT15].

Tab. 3 shows ν for all our tests for rG = 2. Evaluations forrG ∈ {1,3} yielded similar results, omitted here for space reasons.The table shows very good results for tsNET(*): Not only doestsNET(*) often outperform other methods, it also does so by a largemargin. In cases where tsNET(*) is below the median, the marginsare small. Since the metric ν is an inverse measure of the numberof false and missing neighbors, it is related to the observation about

Figure 2: Plots of pairs of input (x-axis) and output (y-axis)distances. Gray value encodes the density of graph-distance vs.Euclidean-distance pairs.

false neighbors in the previous section. This good performance oftsNET(*) is directly related to the neighbor-preserving nature oft-SNE.

4.6. Running time

Tab. 4 gives the running time for tsNET(*) for the larger graphs inour benchmark, using the Python-based implementation, on a 3.4GHz PC running Linux. We see that tsNET* provides a speed-upover tsNET of about 15% on average. Although our Python imple-mentation is not fast, it is simple and easy to understand and use(see also our code [Kru16]). If running time is an important con-sideration, more efficient t-SNE implementations in C, using GPUacceleration [vdM] or quadtrees [vdM14, PLvdM∗15] can be di-rectly used instead. Our method adds little overhead to t-SNE, asthe PivotMDS preprocessor used for tsNET* takes less than a sec-ond on all tested graphs.



tsNET tsNET* IDMAP LSP PMDS SFDP LinLog GRIP NEATOdwt_72 0.855 0.855 0.770 0.676 0.732 0.714 0.692 0.914 0.828lesmis 0.715 0.712 0.748 0.642 0.674 0.729 0.649 0.666 0.695can_96 0.658 0.671 0.535 0.530 0.515 0.547 0.540 0.533 0.565rajat11 0.716 0.717 0.675 0.638 0.624 0.661 0.637 0.634 0.655jazz 0.805 0.804 0.842 0.791 0.827 0.840 0.777 0.824 0.817visbrazil 0.589 0.584 0.476 0.449 0.414 0.471 0.542 0.452 0.425grid17 0.785 0.785 0.812 0.750 0.727 0.751 0.369 0.804 1.000mesh3e1 0.904 0.904 0.993 0.957 0.994 0.809 0.587 0.896 0.999netscience 0.711 0.707 0.539 0.583 0.473 0.622 0.614 0.559 0.510dwt_419 0.739 0.741 0.723 0.741 0.695 0.654 0.542 0.751 0.658price_1000 0.639 0.639 0.483 0.469 0.422 0.528 0.594 0.216 0.284dwt_1005 0.609 0.619 0.512 0.503 0.485 0.523 0.390 0.516 0.455cage8 0.435 0.437 0.207 0.278 0.221 0.235 0.349 0.193 0.200bcsstk09 0.867 0.867 0.767 0.795 0.602 0.835 0.565 0.856 0.973block_2000 0.374 0.372 0.205 0.279 0.166 0.287 0.339 0.155 0.160sierpinski3d 0.579 0.580 0.387 0.492 0.326 0.534 0.438 0.549 0.561CA-GrQc 0.480 0.483 0.170 0.207 0.179 0.183 0.349 0.081 0.119EVA 0.801 0.802 0.707 0.706 0.717 0.696 0.780 0.406 0.4593elt 0.663 0.715 0.415 0.485 0.384 0.595 0.248 0.576 0.506us_powergrid 0.454 0.457 0.234 0.353 0.253 0.429 0.409 0.233 0.215average 0.842 0.852 0.469 0.426 0.296 0.541 0.333 0.386 0.399

Neighborhood preservation (rel.)least preserving (bad) most preserving (good)

mean

Table 3: Neighborhood preservation metric ν for rG = 2 (higher isbetter), indicating how well the input (graph) and output (layout)node-neighborhoods match. Cell colors encode ν on the same row.

tsNET (s) tsNET* (s) tsNET*(% of tsNET)

dwt_72 3.2 3.1 96%

lesmis 4.2 4.7 110%

can_96 3.3 3.6 110%

rajat11 4.3 3.8 89%

jazz 5.1 4.6 91%

visbrazil 10.2 12.4 121%

grid17 5.5 5.3 97%

mesh3e1 5.9 5.5 94%

netscience 9.4 9.3 99%

dwt_419 9.7 7.5 78%

price_1000 182.7 100.8 55%

dwt_1005 63.9 39.4 62%

cage8 59.5 54.0 91%

bcsstk09 35.2 32.9 94%

block_2000 252.7 151.1 60%

sierpinski3d 332.9 206.6 62%

CA-GrQc 911.3 879.3 96%

EVA 1388.3 838.0 60%

3elt 1781.1 1303.1 73%

us_powergrid 1029.9 722.2 70%

Table 4: Running time in seconds of tsNET and tsNET*.

4.7. Bundled layouts

Sect. 4.2 shows that tsNET can retain neighborhoods successfully,at the expense of introducing a few long edges. We can reduce theclutter created by these with the help of edge bundling, [vdZCT16].For this, we bundle the long edges, but keep the short ones un-changed (Fig. 3). This is easily done by modifying any exist-ing general graph-bundling method to enforce a maximal edge-displacement δ as a function of the edge length, where we setδ = 0.25. The result is a ‘hybrid’ graph-drawing in which shortedges are straight lines (as in classical graph drawings) and longedges are bundled, thereby reducing clutter. To our knowledge, thisis the first time that selective bundling has been applied in this wayto declutter the drawing of graphs. This method helps one clearlysee which parts of the graph layout have been ‘torn off’ by tsNETto achieve a globally optimal node placement. Of course, bundlingis also applicable to tsNET* layouts. We chose the tsNET layouts(most notably 3elt) to illustrate this idea as they contained morelong edges that could benefit from bundling.

4.8. 3D layouts

As the formulation of tsNET is independent of the output-space di-mension, it is interesting to study its ability to produce 3D graph

Figure 3: tsNET unbundled (left) and bundled layouts (right) forjazz, cage8, block_2000, and 3elt. Edge colors encode edge lengths((dark) red = shortest, green = median, blue = longest).

layouts. To do this, we consider the problem of recovering geomet-ric information from topological information present in 3D meshes,similar to work presented in [GK01,Wal01]: Given a mesh (Fig. 4,left), we consider the graph G given by its vertices V and cell-edgesE. Next, we use tsNET to create a 3D layout of G (Fig. 4, right).

The tsNET reconstructions preserve the local regular structure ofthe input meshes (Fig. 4, right). This can be attributed to the factthat the underlying t-SNE technique is very well suited to preserveneighbors.

More importantly, we see that tsNET can reconstruct the high-level structure of the input shapes well, see e.g., relative sizes andpositions of the horse’s head and four limbs. However, reconstruc-tion of the exact positions of mesh parts is not possible. This isnot surprising, as Euclidean distances between nodes in the originalmesh are not reflected by their graph-theoretical distances. E.g., theback hooves of the horse are geometrically quite close, but graph-theoretically far away, and will therefore not be placed as closetogether. Also, certain shape parts, such as the horse’s limbs, canbe articulated in the original mesh without significantly modifying



the graph-theoretical distance matrix. These shape parts will be po-sitions by tsNET in ways that may not match their original geomet-rical positions. Finally, let us stress that the aim of this illustrationis not to claim that tsNET can be used as a standalone method for3D geometry reconstruction from topology data; rather, we aim toshow that tsNET can be used to construct 3D graph layouts as eas-ily as for 2D layouts.

Figure 4: Original 3D meshes (left) and meshes reconstructed withtsNET in 3D (right).

5. Discussion

We discuss next several properties of our method.

Ease of use: Our method depends on a single parameter: theperplexity κ, inherited from t-SNE (Sect. 3.2), which was onaverage 80, with a standard deviation of 45, for all our tests, exceptEVA (Tab. 6), where t-SNE needed κ = 600 to converge. Thissensitivity of t-SNE with respect to κ is also documented in otherworks, e.g. [WVJ16]. Apart from this, no other parameters arerequired, which is an advantage against other methods that rely oncareful fine-tuning of multiple parameters.

Determinism: The tsNET algorithm is not deterministic, giventhe random point initialization of t-SNE [vdMH08]. However,tsNET*, which initializes 2D node positions using PivotMDS(Sect. 3), is deterministic, since PivotMDS is so. This importantproperty guarantees that, given the same graph, tsNET* producesthe same layout.

Robustness: We produce good layouts for different types ofgraphs, without fine-tuning parameters, whereas many othermethods we are aware of handle only graphs of a certain type well.

Scalability: It is important to note that speed is not our main goal,but rather the quality of our layouts obtained by the novel use oft-SNE. As mentioned in Sect. 4.6, speed-ups can be obtained withthe help of more advanced t-SNE implementations (at the expenseof code complexity). In fairness, however, we should mention thatsuch speed-ups do not accelerate the computation of the inputdistance matrix, which is O(|V |2).

Generalizability: We currently use the graph’s shortest-path dis-tances to encode the graph information for the layout. More so-phisticated distance metrics can be explored, by incorporating e.g.,betweenness centrality [BG13] in the distance metric, or usingMarkov-chain models of random walks [BG14]. Also, approxima-tions of shortest-path distances can be considered for larger graphswhere computation of the full shortest-path distance matrix is ex-pensive. Finally, given that our layout method is deterministic, itcan be generalized for dynamic graphs, using recent developmentsfor projecting time-dependent data with t-SNE [RFT16].

6. Conclusion

We presented the tsNET and tsNET* methods for computing 2Dlayouts of graphs. Our work shows that modern dimensionality re-duction methods are a good alternative to classical graph layouttechniques. From a practical standpoint, our methods are simpleto implement, and have compact mathematical formulations, lever-aging the proven t-SNE method for dimensionality reduction. Ex-tensive visual and quantitative evaluations show that our methodsresult in excellent quality when compared to state-of-the-art meth-ods.

There are several natural directions for future work. First, ap-proximate versions of t-SNE can be used to significantly accelerateour methods. Second, node and edge weights can be incorporatedin the distance metric to provide a more detailed representation ofthe given graph. Third, 3D layouts as in Fig. 4 can be further in-vestigated. Finally, a direct extension of our methods to computinglayouts of time-dependent graphs can be investigated, given recentdevelopments in dynamic t-SNE [RFT16].

7. Acknowledgements

This work was partly supported by the project MOTO (H2020-SESAR-2015-1), grant 699379, offered by the European Commis-sion.

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tsNET tsNET* IDMAP LSP PMDS SFDP LinLog GRIP NEATOdw

t_72

lesm

isca

n_96

raja

t11

jazz

visb

razi

lgr

id17

mes

h3e1

nets

cien

cedw

t_41

9

Edge lengthshort long

Table 5: Comparison of graph layouts (1/2). Rows and columns correspond to different graphs and layout methods. Edge colors encode theirlengths.



tsNET tsNET* IDMAP LSP PMDS SFDP LinLog GRIP NEATOpr

ice_

1000

dwt_

1005

cage

8bc

sstk

09bl

ock_

2000

sier

pins

ki3d

CA

-GrQ

cE

VA3e

ltus

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ergr

id

Edge lengthshort long

Table 6: Comparison of graph layouts (2/2). Rows and columns correspond to different graphs and layout methods. Edge colors encode theirlengths.



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